hw1 s2 - -k k-2 p k-2(1-p n-k n n-k k p k(1-p n-k = k k-1 n...

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Solution to HW1 Question 4 4. S ( X 1 ,X 2 ) = 1 { X 1 = 1 ,X 2 = 1 } = X 1 X 2 and ES = 1 * P ( X 1 = 1 ,X 2 = 1) + 0 * P ( X 1 X 2 = 0) = p 2 . So S is an unbiased estimator for p 2 . The likelihood function of X 1 ,X 2 , ··· ,X n is: i = n Y i =1 p X i (1 - p ) 1 - X i = p Σ i = n i =1 X i (1 - p ) n - Σ i = n i =1 X i so Σ i = n i =1 X i is a sufficient and complete statistic for p 2 . By Applying Lehman-Scheffe Theorem, the UMVUE for p 2 could be E ( S | Σ i = n i =1 X i ) P ( S = 1 | Σ i = n i =1 X i = k ) = P ( X 1 = 1) P ( X 2 = 1) P i = n i =3 X i = k - 2) P i = n i =1 X i = k ) since Σ i = n i =1 X i B ( n,p ), we have P i = n i =1 X i = k ) = ± n k p k (1 - p ) n - k = n ! ( n - k )! k ! p k (1 - p ) n - k . Similarly, P i = n i =3 X i = k - 2) = ( n - 2)! ( n - k )!( k - 2)! p k - 2 (1 - p ) n - k So P ( S = 1 | Σ i = n i =1 X i = k ) = p * p * ( n - 2)! ( n
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Unformatted text preview: -k )!( k-2)! p k-2 (1-p ) n-k n ! ( n-k )! k ! p k (1-p ) n-k = k ( k-1) n ( n-1) E ( S | Σ i = n i =1 X i = k ) = 1 * P ( S = 1 | Σ i = n i =1 X i = k ) + 0 * P ( S = 0 | Σ i = n i =1 X i = k ) = k ( k-1) n ( n-1) i.e the UMVUE is Σ i = n i =1 X i (Σ i = n i =1 X i-1) n ( n-1) or n ¯ X ( n ¯ X-1) n ( n-1)...
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This homework help was uploaded on 04/20/2008 for the course STATS 131C taught by Professor Polonik during the Spring '07 term at UC Davis.

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